(EDITED 2024/03/11  GMT 17H)

In a recent Question, @cq mentionned in its last reply "In fact, I wanted to do parameter sensitivity analysis and get the functional relationship between the [...] function and [parameters]. Later, i will study how the uncertainty of [the parameters] affects the [...] function".
I did not keep exchanging further on with @cq, simply replying that I could provide it more help if needed.

In a few words the initial problem was this one:

• Let X_1 and X_2 two random variables and G the random variable defined by  G = 1 - (X_1 - 1)^2/9 - (X_2 - 1)^3/16.
   
•  X_1 and X_2 are assumed to be gaussian random variables with respective mean and standard deviation equal to (theta_1, theta_3) and (theta_2, theta_4).
   
• The four theta parameters are themselves assumed to be realizations of four mutually independent uniform random variables Theta_1, ..., Theta_4 whose parameters are constants.
   
• Let QOI  (Quantity Of Interest) denote some scalar statistic of G (for instance its Mean, Variance, Skweness, ...).
  For instance, if QOI = Mean(G), then  QOI expresses itself as a function of the four parameters theta_1, ..., theta_4.
  The goal of @cq is to understand which of those parameters have the greatest influence on QOI.

For a quick survey of Sensitivity Analysys (SA) and a presentation of some of the most common strategies see Wiki-Overview


The simplest SA is the Local SA (LSA) we are all taught at school: having chosen some reference point P in the [theta_1, ..., theta_4] space the 1st order partial derivatives d[n] = diff(QOI, theta_n) expressed at point P give a "measure" (maybe after some normalization) of the sensitivity, at point P, of QOI regardibg each parameter theta_n.


A more interesting situation occurs when the parameters can take values in a neighorhood of  P which is not infinitesimal, or more generally in some domain without reference to any specific point P.
That is where Global SA (GSA) comes into the picture.
While the notion of local variation at some point P is well established, GSA raises the fundamental question of how to define how to measure the variation of a function over an arbitrary domain?
Let us take a very simple example while trying to answer this question "What is the variation of sin(x) over [0, 2*Pi]?"

1. If we focus on the global trend of sin(x)  mean there is no variation at all.
2. If we consider peak-to-peak amplitude the variation is equal to 2.
3. At last, if we consider L2 norm the variation is equal to Pi.
   (but the constant function x -> A/sqrt(2) has the same L2 norm but it is flat, and in some sense les fluctuating). 

Statisticians are accustomed to use the concept of variance as a measure to quantify the dispersion of a random variable. At the end of the sixties  one of them, Ilya Meyerovich Sobol’,  introduced the notion of Variance-Based GSA as the key tool to define the global variation of a function. This notion naturally led to that of Sobol' indices as a measure of the sensitivity of a funcion regarding one of its parameters or, which most important, regarding any combination (on says interaction) of its parameters.

The aim of this post is to show how Sobol' indices can be computed when the function under study has an analytic expression.
 

The Sobol' analysis is based on an additive decomposition of this function in terms of 2^P mutually orthogonal fiunctions where P is the number of its random parameters.
This decomposition and the ensuing integrations whose values will represent the Sobol' indices can be done analytically in some situation. When it is no longer the case specific numerical estimation methods have to be used;


The attached file contains a quite generic procedure to compute exact Sobol' indices and total Sobol' indices for a function whose parameters have any arbitrary statistical distribution.
Let's immediately put this into perspective by saying that these calculations are only possible if Maple is capable to find closed form expressions of some integrals, which is of course not always the case.

A few examples are also provided, including the one corresponding to @cq's original question.
At last two numerical estimation methods are presented.

SOBOL.mw